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Artificially cemented soft soil samples were produced by mixing kaolin, beta hemihydrate and water. Oedometer and triaxial tests were performed on this material, which was shown to demonstrate many of the characteristics of natural soft clays. Tests were performed on the material at a range of consolidation pressures and at different strain rates. It was shown that the shear strength behaviour of the material exhibits complex characteristics that require explicit consideration of soil structure at the loading stage. Similarly to other natural clays, the critical state could not be achieved in the triaxial apparatus; rather, it was shown that a post-rupture state was achieved which appears remote from the intrinsic critical state. Different normalisation techniques were compared. It is recommended that normalisation of behaviour should be based on the virgin consolidation line as it defines a space in which the soil can only exist due to structure.

The ongoing study deals with various forms of solutions for the biological population model with a novel beta-time derivative operators. This model is very conducive to explain the enlargement of viruses, parasites and diseases. This configuration of the aforesaid classical scheme is scouted for its new solutions especially in soliton shape via two of the well known analytical strategies, namely: the extended Sinh-Gordon equation expansion method (EShGEEM) and the Expa function method. These soliton solutions suggest that these methods have widened the scope for generating solitary waves and other solutions of fractional differential equations. Different types of soliton solutions will be gained such as dark, bright and singular solitons solutions with certain conditions. Furthermore, the obtained results can also be used in describing the biological population model in some better way. The numerical solution for the model is obtained using the finite difference method. The numerical simulations of some selected results are also given through their physical explanations. To the best of our knowledge, No previous literature discussed this model through the application of the EShGEEM and the Expa function method and supported their new obtained results by numerical analysis.

In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $. The derivation is extended from a recently published paper by Huseynov et al. in [1], which is limited for incommensurate fractional order $ 0 < \alpha, \beta < 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 < \alpha, \beta < 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.

The iterative functional equation $ \alpha\rightarrow(\alpha\rightarrow \beta) = \alpha\rightarrow \beta $ and the law of importation $ (\alpha\wedge \beta)\rightarrow \gamma = \alpha\rightarrow (\beta\rightarrow \gamma) $ are two tautologies in classical logic. In fuzzy logics, they are two important properties, and are respectively formulated as $ I(\alpha, \beta) = I(\alpha, I(\alpha, \beta)) $ and $ I(T(\alpha, \beta), \gamma) = I(\alpha, I(\beta, \gamma)) $ where $ I $ is a fuzzy implication and $ T $ is a $ t $-norm. Over the past several years, solutions to these two functional equations involving different classes of fuzzy implications have been studied. However, there are no results about stability study of fuzzy functional equations involving fuzzy implication. This paper discusses fuzzy implications that do not strictly satisfying these equations, but approximately satisfy these equations. Then we establish the Hyers-Ulam stability of the iterative functional equation involving the $ (S, N) $-implication, where the $ (S, N) $-implication is a common class of fuzzy implications generated by a continuous $ t $-conorm $ S $ and a continuous fuzzy negation $ N $. Furthermore, given a fixed $ t $-norm (the minimum $ t $-norm or the product $ t $-norm) the Hyers-Ulam stability of the law of importation involving the $ (S, N) $-implication is studied.

We show that the weak disorder phase for the directed polymer model in a random environment is characterized by the integrability of the running supremum $$\sup _{n\in \mathbb N}W_n^\beta $$ of the associated martingale $$(W_n^\beta )_{n\in \mathbb N}$$ . If the environment is bounded, we also show that $$(W_n^\beta )_{n\in \mathbb N}$$ is $$L^p$$ -bounded in the whole weak disorder phase, for some $$p>1$$ . The argument generalizes to non-negative martingales with a certain product structure.

The main motive of this study is to present a new class of a generalized k -Bessel–Maitland function by utilizing the k -gamma function and Pochhammer k -symbol. By this approach, we deduce a few analytical properties as usual differentiations and integral transforms (likewise, Laplace transform, Whittaker transform, beta transform, and so forth) for our presented k -Bessel–Maitland function. Also, the k -fractional integration and k -fractional differentiation of abovementioned k -Bessel–Maitland functions are also pointed out systematically.

A new BEME distribution known as beta Exponentiated moment exponential (BEME) distribution is proposed. We provide here some shape properties, moments in the form of special functions, mean deviations of BEME distribution. We derive mathematical properties of the BEME distribution including the reliability measures, the Bonferroni and the Lorenz curves, rth order statistics, measures of uncertainty: the Shannon entropy measure and the s-entropy measure. The parameters of the BEME distribution are estimated by the method of maximum likelihood estimation and estimated non-linear equations for these estimates are presented. The application of BEME distribution is explored in three different fields of engineering.

The SKKUCY-10 cyclotron based on 83.2 MHz, 40 kV half-wave RF cavity was developed at Sungkyunkwan University for the production of medical radioisotopes. The resonant frequency $$f_{{{\text{RF}}}}$$ of the cyclotron and the RF coupling coefficient $$\beta _{{\text{c}}}$$ of the RF cavity system were measured at various vacuum, and temperature conditions. The normalized multi-pacting intensities at four positions in the power coupler were analyzed to predict the multi-pacting power. Differences, $$\Delta f_{{{\text{RF}}}}$$ and $$\Delta \beta _{{\text{c}}}$$ , caused by the vacuum, and temperature conditions were modified based on the coupler and tuner gap distances. During the RF conditioning, a constant 15 kW pulse mode and a variable 1 to 15 kW continuous wave mode were employed. The values of the reflection coefficient $$\Gamma$$ and $$\beta _{{\text{c}}}$$ were 1.2% and 0.8, respectively, when the cavity dissipation power was 12.4 kW at 83.2 MHz. Good agreement between the simulation and experimental data was obtained.

The General Project of Science and Technology Plan of Beijing Municipal Commission of Education. Grant Number: KM201910009014; Shanghai Planning Office of Philosophy and Social Science. Grant Number: 2019EGL018; The Fundamental Research Funds for the Central Universities. Grant Number: 2020RC38; Key Technologies R & D Program of He'nan Province. Grant Number: 212102210084; The National Natural Science Foundation of China (NSFC). Grant Number: 71942003.

We consider a many-body Bosonic system with pairwise particle interaction given by $$N^{3\beta -1}v(N^\beta x)$$ where $$0

The article describes the modern software for calculation of protection against bremsstrahlung of beta-sources of different compound and combinations of radiation shield materials. The description and general opportunities of software are depicted. It was presented during the validation procedure that attained results of software work are in good agreement with reference values.

We study the Reconstruction-of-the-Measure Problem (ROMP) for commuting 2-variable weighted shifts $$W_{(\alpha ,\beta )}$$ , when the initial data are given as the Berger measure of the restriction of $$W_{(\alpha ,\beta )}$$ to a canonical invariant subspace, together with the marginal measures for the 0–th row and 0–th column in the weight diagram for $$W_{(\alpha ,\beta )}$$ . We prove that the natural necessary conditions are indeed sufficient. When the initial data correspond to a soluble problem, we give a concrete formula for the Berger measure of $$W_{(\alpha ,\beta )}$$ . Our strategy is to build on previous results for back-step extensions and one-step extensions. A key new theorem allows us to solve ROMP for two-step extensions. This, in turn, leads to a solution of ROMP for arbitrary canonical invariant subspaces of $$\ell ^2({\mathbb {Z}}_+^2)$$ .

Suppose $$\{P_{n}^{(\alpha , \beta )}(x)\} _{n=0}^\infty $$ is a sequence of Jacobi polynomials with $$ \alpha , \beta >-1.$$ We discuss special cases of a question raised by Alan Sokal at OPSFA in 2019, namely, whether the zeros of $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n+k}^{(\alpha + t, \beta + s )}(x)$$ are interlacing if $$s,t >0$$ and $$ k \in {\mathbb {N}}.$$ We consider two cases of this question for Jacobi polynomials of consecutive degree and prove that the zeros of $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n+1}^{(\alpha , \beta + 1 )}(x),$$ $$ \alpha> -1, \beta > 0, $$ $$ n \in {\mathbb {N}},$$ are partially, but in general not fully, interlacing depending on the values of $$\alpha , \beta $$ and n. A similar result holds for the extent to which interlacing holds between the zeros of $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n+1}^{(\alpha + 1, \beta + 1 )}(x),$$ $$ \alpha>-1, \beta > -1.$$ It is known that the zeros of the equal degree Jacobi polynomials $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n}^{(\alpha - t, \beta + s )}(x)$$ are interlacing for $$ \alpha -t> -1, \beta > -1, $$ $$0 \le t,s \le 2.$$ We prove that partial, but in general not full, interlacing of zeros holds between the zeros of $$ P_{n}^{(\alpha ,\beta )}(x)$$ and $$ P_{n}^{(\alpha + 1, \beta + 1 )}(x),$$ when $$ \alpha> -1, \beta > -1.$$ We provide numerical examples that confirm that the results we prove cannot be strengthened in general. The symmetric case $$\alpha = \beta = \lambda -1/2$$ of the Jacobi polynomials is also considered. We prove that the zeros of the ultraspherical polynomials $$ C_{n}^{(\lambda )}(x)$$ and $$ C_{n + 1}^{(\lambda +1)}(x),$$ $$ \lambda > -1/2,$$ are partially, but in general not fully, interlacing. The interlacing of the zeros of the equal degree ultraspherical polynomials $$ C_{n}^{(\lambda )}(x)$$ and $$ C_{n}^{(\lambda +3)}(x),$$ $$ \lambda > -1/2,$$ is also discussed.

We investigate joint temporal and contemporaneous aggregation of N independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient $\alpha\in(0,1)$ and with idiosyncratic Poisson innovations. Assuming that $\alpha$ has a density function of the form $\psi(x) (1 - x)^\beta$, $x \in (0,1)$, with $\beta\in(-1,\infty)$ and $\lim_{x\uparrow 1} \psi(x) = \psi_1 \in (0,\infty)$, different limits of appropriately centered and scaled aggregated partial sums are shown to exist for $\beta\in(-1,0]$ in the so-called simultaneous case, i.e., when both $N$ and the time scale $n$ increase to infinity at a given rate. The case $\beta\in(0,\infty)$ remains open. We also give a new explicit formula for the joint characteristic functions of finite dimensional distributions of the appropriately centered aggregated process in question., Comment: 38 pages

"In this paper, we introduce new class $\Sigma_{m}(\mu,\lambda,\gamma,\beta)$ of $m$-fold symmetric bi-univalent functions. Furthermore, we use the Faber polynomial expansions to find upper bounds for the general coefficients $|a_{mk+1}|(k \geqq 2)$ of functions in the class $\Sigma_{m}(\mu,\lambda,\gamma,\beta)$. Moreover, we obtain estimates for the initial coefficients $|a_{m+1}| $ and $|a_{2m+1}|$ for functions in this class. The results presented in this paper would generalize and improve some recent works."

There are many extensions and generalizations of Gamma and Beta functions in the literature. However, a new extension of the extended Beta function B_(ζ〖, α〗_1)^(α_2;〖 m〗_1,〖 m〗_2 ) (a_1,a_2 ) was introduced and presented here because of its important properties. The new extended Beta function has symmetric property, integral representations, Mellin transform, inverse Mellin transform and statistical properties like Beta distribution, mean, variance, moment and cumulative distribution which ware also presented. Finally, the new extended Gauss and Confluent Hypergeometric functions with their propertied were introduced and presented.

Current interpretations of the LHC results on two Higgs doublet models (2HDM) underestimate the sensitivity due to neglecting higher order effects. In this work, we revisit the impact of these effects using the current cross-section times branching ratio limits of the $A\to hZ, H \to VV$ and $H\to hh$ channels. With a degenerate heavy Higgs mass $m_\Phi$, we find that the LHC searches gain sensitivity to the small $\tan\beta$ region after including loop corrections, even close to $\cos(\beta-\alpha)=0$ which is not reachable at tree level for all types of 2HDM. For a benchmark point with $m_\Phi=300$ GeV, $\tan\beta, Comment: 15 pages

In this article, we deal with the solutions of the difference analogue of Fermat-type equation of the form $$f^{3}(z)+[c_{1}f(z+c)+c_{0}f(z)]^{3}=e^{\alpha z+\beta}$$ and prove a result generalizing a result of Han and Lu [J. Contemp. Math. Anal. 2019] and Ma et al. [J. Funct. Spaces 2020]. Furthermore, we explore the class of functions satisfying the Fermat-type difference equation. A considerable number of examples have been exhibited throughout the paper pertinent with the different issues. We characterize all possible nonconstant solutions of the Fermat-type difference equation $$f^{2}(z)+f^{2}(z+c)=e^{\alpha z+\beta}$$ .

We study the relationship between the existence of an $$\upsilon$$ -density $$\Delta^\gamma (\alpha,\beta)$$ of roots along the curves of regular rotation of an entire function of zero order and the regular behavior at infinity of its logarithm.

As is well known, the definitions of fractional sum and fractional difference of f (z) on non-uniform lattices x(z) = c1z2 + c2z + c3 or x(z) = c1qz + c2q−z + c3 are more difficult and complicated. In this article, for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways. The analogue of Euler’s Beta formula, Cauchy’ Beta formula on non-uniform lattices are established, and some fundamental theorems of fractional calculas, the solution of the generalized Abel equation on non-uniform lattices are obtained etc.

The minimum-variance portfolio (MVP) has become an essential part of modern portfolio theory, largely due to the availability of its analytical formula and its good out-of-sample performance. When extra constraints such as the long-only constraints are added, MVP in general does not admit an analytical formula anymore. An exceptional case is when the single-factor model holds among the securities considered. It is known that there exists a semi-closed form formula, which relies on an unknown quantity called the long-only beta threshold ( β L ). This note conducts a detailed study of β L and establishes its optimality property that it is the smallest among all possible thresholds. We also develop a simple bisection method for computing β L . Finally we include a mathematical proof for this semi-closed form formula. The results reported provide a deep understanding how the long-only MVP are constructed.

An outer independent (double) Roman dominating function is a (double) Roman dominating function f for which the set of vertices assigned 0 under f is independent. The outer independent (double) Roman domination number ( $$\gamma _{oidR}(G)$$ ) $$\gamma _{oiR}(G)$$ is the minimum weight taken over all outer independent (double) Roman dominating functions of G. A vertex cover number $$\beta (G)$$ is the minimum size of any vertex cover sets of a graph G. In this work, we present some contributions to the study of outer independent double Roman domination in graphs. Characterizations of the families of all connected graphs with small outer independent double Roman domination numbers, and tight lower and upper bounds on this parameter are given. We also prove that the decision problem associated with $$\gamma _{oidR}(G)$$ is NP-complete even when restricted to planar graphs with maximum degree at most four. We moreover prove that $$2\beta (T)+1\le \gamma _{oidR}(T)\le 3\beta (T)$$ for any tree T, and show that each integer between the lower and upper bounds is realizable. Finally, we give an exact formula for this parameter concerning the corona graphs.

It is shown that the maximal operator of some (C, βn)-means of cubical partial sums of two variable Walsh–Fourier series of integrable functions is of weak type (L1,L1). Moreover, the (C, βn)-means $$ {\sigma}_{2^n}^{\beta_n}f $$ of the function f ∈ L1 converge a.e. to f for f ∈ L1(I2), where I is the Walsh group for some sequences 1 > βn ↘ 0.

Coincidence counting (CC) is a widely applied method for radionuclide activity measurement. The $4\pi \beta ({\mathrm{ LS}}) -\gamma $ coincidence counting is a type of CC that is applied to determine the activity of radionuclides that undergo $\beta -\gamma $ decay. A liquid scintillation (LS) counter has the advantages of no self-absorption and simple sample preparation. Hence, this study focuses on the $4\pi \beta ({\mathrm{ LS}}) -\gamma $ method to measure the activity of LS sources with a standard triple-to-double coincidence ratio (TDCR) system for LS measurements in the $\beta $ channel and a NaI(Tl) scintillation detector in the $\gamma $ channel. In this study, high-accuracy digital coincidence counting (DCC) software was developed to determine the activity of radionuclides using this $4\pi \beta ({\mathrm{ LS}}) -\gamma $ coincidence counting method. The software is characterized by the application of the living time method during true event discrimination and the TDCR technology to estimate the effect of system noise and after-pulses in the $\beta $ channel. The developed software was validated by comparing the results obtained using the software with standardized $^{60}\text {Co}$ activity measurement at the National Institute of Metrology (NIM) applying previously validated methods. The results showed that the activity measured by the developed DCC software had a tolerable error. It has good potential activity measurement, and it is expected to be applied to calibrate radionuclides with more complex decay diagrams in the future.

This paper studies the Cauchy problem of the 3D incompressible micropolar equations with a damping term $\sigma |u|^{\beta -1}u$ σ | u | β − 1 u ($\sigma >0, 1\le \beta σ > 0 , 1 ≤ β < 3 ). It is shown that the strong solutions exist globally for any $1\le \beta 1 ≤ β < 3 .

Non-negative latent factor (NLF) models well represent high-dimensional and sparse (HiDS) matrices filled with non-negative data, which are frequently encountered in industrial applications like recommender systems. However, current NLF models mostly adopt Euclidean distance in their objective function, which represents a special case of a ${\boldsymbol{\beta }}$ -divergence function. Hence, it is highly desired to design a ${\boldsymbol{\beta }}$ -divergence-based NLF ( ${\boldsymbol{\beta }}$ -NLF) model that uses a ${\boldsymbol{\beta }}$ -divergence function, and investigate its performance in recommender systems as ${\boldsymbol{\beta }}$ varies. To do so, we first model ${\boldsymbol{\beta }}$ -NLF’s learning objective with a ${\boldsymbol{\beta }}$ -divergence function. Subsequently, we deduce a general single latent factor-dependent, non-negative and multiplicative update scheme for ${\boldsymbol{\beta }}$ -NLF, and then design an efficient ${\boldsymbol{\beta }}$ -NLF algorithm. The experimental results on HiDS matrices from industrial applications indicate that by carefully choosing the value of ${\boldsymbol{\beta }}$ , ${\boldsymbol{\beta }}$ -NLF outperforms an NLF model with Euclidean distance in terms of accuracy for missing data prediction without increasing computational time. The research outcomes show the necessity of using an optimal ${\boldsymbol{\beta }}$ -divergence function in order to achieve the best performance of an NLF model on HiDS matrices. Hence, the proposed model has both theoretical and application significance.

Let $$a_{j, 2j}(n)$$ denote the number of [j, 2j]-cubic partitions of n in which none of the parts congruent to j modulo 2j. In this paper, we have establish many infinite families of congruences modulo 3 for $$a_{3, 6}(n)$$ , congruences modulo powers of 2 for $$a_{5, 10}(n)$$ and congruences modulo powers of 2 and 3 for $$a_{9, 18}(n).$$ For example, for all $$n \ge 0$$ and $$\beta , \gamma \ge 0,$$ $$\begin{aligned} {a}_{9, 18}\left(162\cdot 5^{4\beta }\cdot 7^{2\gamma +1}(7n+j)+\dfrac{27\cdot 5^{4\beta }\cdot 7^{2\gamma +2}+1}{2}\right) \equiv 0 \pmod {27}, \end{aligned}$$ where $$j=1, 2, 3, 4, 5, 6.$$

We find radial and nonradial solutions to the following nonlocal problem $$-\Delta u +\omega u= \big(I_\alpha\ast F(u)\big)f(u)-\big(I_\beta\ast G(u)\big)g(u) \text{ in } \mathbb{R}^N$$ under general assumptions, in the spirit of Berestycki and Lions, imposed on $f$ and $g$, where $N\geq 3$, $0\leq \beta \leq \alpha0$, then we deal with two competing nonlocal terms modelling attractive and repulsive interaction potentials., Comment: 18 pages, to appear in Nonlinearity

The aim of this paper is to fill in the gaps in the formulation and the proof of a theorem contained in the paper by K.Ozeki (Aequ Math 25:247–252, 1982) published in this journal. We also give a short proof of this theorem and use it to obtain certain information about the factorization of polynomials of the form $$f(x)-f(y)$$ .

In this article, we define two new families $${\mathcal{T}}_{\Sigma }(\mu ,\gamma ,\lambda ;\alpha )$$ and $${\mathcal{T}}_{\Sigma }^{*}(\mu ,\gamma ,\lambda ;\beta )$$ of normalized holomorphic and bi-univalent functions which involve the Bazilevic functions and the $$\lambda$$ -pseudo-starlike functions. For functions in each of these families, we establish the bounds for $$|a_2|$$ and $$|a_3|$$ , where $$a_2$$ and $$a_3$$ are the initial Taylor-Maclaurin coefficients. We also point out several special cases and consequences of our results.

We deal with an alienation problem for an Euler–Lagrange type functional equation $$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) = 2\alpha ^2f(x) + 2\beta ^2f(y) \end{aligned}$$ f ( α x + β y ) + f ( α x - β y ) = 2 α 2 f ( x ) + 2 β 2 f ( y ) assumed for fixed nonzero real numbers $$\alpha ,\beta ,\, 1 \ne \alpha ^2 \ne \beta ^2$$ α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation $$\begin{aligned} g(x+y) + g(x-y) = 2g(x) + 2g(y). \end{aligned}$$ g ( x + y ) + g ( x - y ) = 2 g ( x ) + 2 g ( y ) . We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case $$g = \gamma f$$ g = γ f was examined.

UDC 517.5 Recently, Kumar et al. proposed a conjecture concerning the convolution of a generalized right half-plane mapping with a vertical strip mapping. They have verified the above conjecture for $n=1,2,3$ and $4$. Also, it has been proved only for $\beta=\pi/2$. In this paper, by using of a new method, we settle this conjecture in the affirmative for all $n\in\mathbb{N}$ and $\beta\in(0,\pi)$. Moreover, we will use this method to prove some results on convolution of harmonic mappings. This new method simplifies calculations and shortens the proof of results remarkably.

Let $C(L)$ be the ring of all continuous real functions on a frame $L$ and $S\subseteq{\mathbb R}$. An $\alpha\in C(L)$ is said to be an overlap of $S$, denoted by $\alpha\blacktriangleleft S$, whenever $u\cap S\subseteq v\cap S$ implies $\alpha(u)\leq\alpha(v)$ for every open sets $u$ and $v$ in $\mathbb{R}$. This concept was first introduced by A. Karimi-Feizabadi, A.A. Estaji, M. Robat-Sarpoushi in {\it Pointfree version of image of real-valued continuous functions} (2018). Although this concept is a suitable model for their purpose, it ultimately does not provide a clear definition of the range of continuous functions in the context of pointfree topology. In this paper, we will introduce a concept which is called pre-image, denoted by ${\rm pim}$, as a pointfree version of the image of real-valued continuous functions on a topological space $X$. We investigate this concept and in addition to showing ${\rm pim}(\alpha)=\bigcap\{S\subseteq{\mathbb R}:~\alpha\blacktriangleleft S\}$, we will see that this concept is a good surrogate for the image of continuous real functions. For instance, we prove, under some achievable conditions, we have ${\rm pim}(\alpha\vee\beta)\subseteq {\rm pim}(\alpha)\cup {\rm pim}(\beta)$, ${\rm pim}(\alpha\wedge\beta)\subseteq {\rm pim}(\alpha)\cap {\rm pim}(\beta)$, ${\rm pim}(\alpha\beta)\subseteq {\rm pim}(\alpha){\rm pim}(\beta)$ and ${\rm pim}(\alpha+\beta)\subseteq {\rm pim}(\alpha)+{\rm pim}(\beta)$.

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrodinger equation $$i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u$$ in $$H^1$$ . The well-posedness theory in $$H^1$$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $$\beta =(4-2\alpha )/(n-2)$$ . It is still an open problem. The main contribution of this paper is to develop the theory in this case.

Let $(X, \tau)$ and $(Y, \sigma)$ be topological spaces in which no separation axioms are assumed, unless explicitly stated and if $\mathcal{I}$ is an ideal on $X$.Given a multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$, $\alpha,\beta$ operators on $(X, \tau)$, $\theta,\delta$ operators on $(Y, \sigma)$ and $\mathcal{I}$ a proper ideal on $X$. We introduce and study upper and lower $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous multifunctions.A multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$ is said to be: {1)} upper-$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous if $\alpha(F^{+}(\delta(V)))\setminus \beta(F^{+}(\theta(V)))\in \mathcal{I}$ for each open subset $V$ of $Y$;\{2)} lower-$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous if$\alpha(F^{-}(\delta(V)))\setminus \beta(F^{-}(\theta(V)))\in \mathcal{I}$ for each open subset $V$ of $Y$;\ {3)} $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous if it is upper-\ %$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuousand lower-$(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous. In particular, the following statements are proved in the article (Theorem 2):Let $\alpha,\beta$ be operators on $(X, \tau)$ and $\theta, \theta^{*}, \delta$ operators on $(Y, \sigma)$: \noi\ \ {1.} The multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$ is upper $(\alpha,\beta,\theta\cap \theta^{*},\delta,\mathcal{I})$-continuous if and only if it is both upper $(\alpha,\beta,\theta,\delta,\mathcal{I})$-continuous and upper $(\alpha,\beta,\theta^{*},\delta,\mathcal{I})$-continuous. \noi\ \ {2.} The multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$ is lower $(\alpha,\beta,\theta\cap \theta^{*},\delta,\mathcal{I})$-continuous if and only if it is both lower $(\alpha,\beta,\theta,\delta,\mathcal{I})$-continuous and lower $(\alpha,\beta,\theta^{*},\delta,\mathcal{I})$-continuous,provided that $\beta(A\cap B) =\beta(A)\cap \beta(B)$ for any subset $A,B$ of $X$.

We study the geometry of sets based on the behavior of the Jones function, \(J_{E}(x) = \int_{0}^{1} \beta_{E;2}^{1}(x,r)^{2} \frac{dr}{r}\). We construct two examples of countably 1-rectifiable sets in \(\mathbf{R}^{2}\) with positive and finite \(\mathcal{H}^1\)-measure for which the Jones function is nowhere locally integrable. These examples satisfy different regularity properties: one is connected and one is Ahlfors regular. Both examples can be generalized to higher-dimension and co-dimension.

In this paper we introduce $$B_{\alpha ,\beta }^{k}$$ -manifolds as generalizations of the notion of smooth manifolds with G-structure or with k-bounded geometry. These are $$C^{k}$$ -manifolds whose transition functions $$\varphi _{ji}=\varphi _{j}\circ \varphi _{i}^{-1}$$ are such that $$\partial ^{\mu }\varphi _{ji}\in B_{\alpha (r)}\cap C^{k-\beta (r)}$$ for every $$\vert \mu \vert =r$$ , where $$B=(B_{r})_{r\in \Gamma }$$ is some sequence of presheaves of Frechet spaces endowed with further structures, $$\Gamma \subset {\mathbb {Z}}_{\ge 0}$$ is some parameter set and $$\alpha ,\beta$$ are functions. We present embedding theorems for the presheaf category of those structural presheaves B. The existence problem of $$B_{\alpha ,\beta }^{k}$$ -structures on $$C^{k}$$ -manifolds is studied and it is proved that under certain conditions on B, $$\alpha$$ and $$\beta$$ , the forgetful functor from $$C^{k}$$ -manifolds to $$B_{\alpha ,\beta }^{k}$$ -manifolds has adjoints.

In this paper, we study three-dimensional equatorial flows with constant vorticity beneath a wave train and above a flat bed in the modified $$\beta $$ -plane approximation. It is proved that the equatorial flow is necessarily irrotational and the free surface is necessarily flat if it exhibits a constant vorticity, due to the incorporation of a gravitational-correction term in the tangent plane approximation.

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts [Formula: see text] and its relation to direct topological factorizations. We show that any reversible CA [Formula: see text] has an almost equicontinuous direction whenever [Formula: see text] is not sofic. This has the corollary that non-sofic beta-shifts are topologically direct prime, i.e. they are not conjugate to direct topological factorizations [Formula: see text] of two nontrivial subshifts [Formula: see text] and [Formula: see text]. We also give a simple criterion to determine whether [Formula: see text] is conjugate to [Formula: see text] for a given integer [Formula: see text] and a given real [Formula: see text] when [Formula: see text] is a subshift of finite type. When [Formula: see text] is strictly sofic, we show that such a conjugacy is not possible at least when [Formula: see text] is a quadratic Pisot number of degree [Formula: see text]. We conclude by using direct factorizations to give a new proof for the classification of reversible multiplication automata on beta-shifts with integral base and ask whether nontrivial multiplication automata exist when the base is not an integer.

Gene transcription in single cells is inherently a probabilistic process. The relationship between variance ($ \sigma^{2} $) and mean expression ($ \mu $) is of paramount importance for investigations into the evolutionary origins and consequences of noise in gene expression. It is often formulated as $ \log \left({{{\sigma}^{2}}}/{{{\mu}^{2}}}\; \right) = \beta\log\mu+\log\alpha $, where $ \beta $ is a key parameter since its sign determines the qualitative dependence of noise on mean. We reveal that the sign of $ \beta $ is controlled completely by external regulation, but independent of promoter structure. Specifically, it is negative if regulators as stochastic variables are independent but positive if they are correlated. The essential mechanism revealed here can well interpret diverse experimental phenomena underlying expression noise. Our results imply that external regulation rather than promoter sequence governs the mean-noise relationship.

In this work, we apply generalized Saigo fractional differential and integral operators having $ k $-hypergeometric function as a kernel, to extended Lommel-Wright function. The results are communicated in the form of the k-Wright function and are utilized to compute beta transform. The novelty and the generalization of the obtained results are shown by relating them with existing literature as special cases.

The beta and gamma functions have recently seen several developments and various extensions because of their nice properties and interesting applications. The contribution of this paper falls within this framework. After introducing a generalized gamma function and two generalized beta functions in several variables, we investigate some inequalities involving these generalized functions.

In this paper, we are interested in a viscoelastic Moore–Gibson–Thompson equation with a type-II memory term and a relaxation function satisfying $$g^{\prime }(t)\le -\eta (t)g(t)$$ . By constructing appropriate Lyapunov functionals in the Fourier space, we establish a general decay estimate of the solution under the condition $$\left( \beta -\frac{\gamma }{\alpha }-\frac{\varrho }{2}\right) >0.$$ We then give the decay rate of the L $$^{2}$$ -norm of the solution. We also give two examples to illustrate our theoretical results.

In this manuscript, we initiate the concept of rectangular $\alpha$-$G$-admissible mappings with respect to $\beta$ and we consider related type contractions in the setting of $G$-metric spaces. We provide some fixed point results. Also, some examples are given to support the obtained results.

The characteristic of count data that have a high frequency of zeros and ones can be considered under a zero-one inflated distribution. In this article, we present a zero-one inflated negative bino...

This article extends the semidiscrete maximal $L^p$-regularity results in Li (2019, Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp., 88, 1--44) to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+\beta }$, where $d$ is the dimension of space and $\beta>0$. The maximal angles of $R$-boundedness are characterized for the analytic semigroup $e^{zA_h}$ and the resolvent operator $z(z-A_h)^{-1}$, respectively, associated to an elliptic finite element operator $A_h$. Maximal $L^p$-regularity, an optimal $\ell ^p(L^q)$ error estimate and an $\ell ^p(W^{1,q})$ estimate are established for fully discrete finite element methods with multistep backward differentiation formulae.